On the passivity properties of a new family of repetitive (hyperbolic) controllers

Transcription

1 Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 8 On the passivity properties of a new family of repetitive hyperbolic controllers P.G. Hernández-Briones, G. Escobar and R. Ortega Abstract This paper studies the passivity properties of three recently reported repetitive schemes [1], []. They are referred as negative, positive and 6l ± 1 repetitive compensators. The first two controllers are composed of a array of a single delay line, while the third controller comprises the array of two delay lines. As most repetitive schemes, these three schemes are intended for the compensation or tracking of period signals, which are composed of harmonic components of a fundamental frequency. In particular, the negative scheme is aimed for the compensation of odd-harmonic components, the positive scheme for the compensation of all harmonics, and the 6l±1 scheme for the compensation of 6l±1 l, 1,,..., harmonics. It is shown here that all three schemes have also equivalent expressions in terms of hyperbolic functions. The main contribution of the present work is to show that these three schemes are discrete-time positive and thus passive. Moreover, it is shown that, after a a modification, motivated by practical issues, these schemes become strictly passive. I. INTRODUCTION Recently, two novel repetitive schemes have been proposed in [1] aimed to compensate for selected harmonics of periodic disturbances. These schemes, in contrast to conventional repetitive schemes [3]-[7], include a feedforward path aimed to enhance their selectivity. Moreover, it was also introduced a negative scheme, which differs from the usual positive used in conventional schemes. A similar scheme using negative can also be found in [8]. An interesting observation in the present work is that these schemes, owning such a particular structure, have equivalent representations in terms of hyperbolic functions. In particular, the negative plus the feedforward scheme can be expressed as a hyperbolic tangent function, while the positive scheme can be expressed as a hyperbolic cotangent function. In [] a new scheme aimed for the compensation of the 6l ± 1l, 1,,..., harmonics has been proposed. This scheme is composed of a array of two delay lines plus a feedforward path. As in the previous cases, this scheme has also an equivalent representation in terms of hyperbolic functions. As most repetitive schemes, the three schemes studied here, are intended for the tracking or rejection of periodic signals, that is, for the compensation of harmonic components of the fundamental frequency, which is referred along the paper P.G. Hernández-Briones and G. Escobar are with the Lab of Proc. and Quality of Energy, IPICYT, San Luis Potosí, SLP 7816, México. [perla,gescobar]@ipicyt.edu.mx R. Ortega is with LSS/CNRS, SUPELEC, Gif-sur-Yvette, Paris 9119, France. Romeo.Ortega@lss.supelec.fr as ω. It has been shown in [1] that, based on the fact that the negative hyperbolic tangent scheme creates poles at odd multiples of the fundamental, then it is able to compensate odd harmonics only see also [8]. It is shown also that the positive scheme hyperbolic cotangent creates poles at every single multiple of the fundamental frequency, therefore, it is able to compensate every even and odd harmonics. On the other hand, it is shown in [] that the third repetitive scheme, creates poles at every 6l ± 1l, 1,,..., multiples of ω,and thus, it is able to compensate the 6l ± 1l, 1,,... harmonics only. This scheme is of particular interest in industrial applications where many processes involve the use of six pulse converters producing harmonics at multiples 6l ± 1l, 1,,... of the fundamental line frequency [13]. It has been shown also that the introduction of the feedforward path in the above schemes creates an infinite set of zeros which are located between every two consecutive poles. The last has the advantage of enhancing the controller selectivity, a characteristic very well appreciated in harmonic distortion compensation. This paper studies the passivity properties of these three schemes, each scheme is studied in a separate section. It is shown that all of three scheme are discrete-time positive PR and thus passive. Moreover, it is shown that all these scheme become strictly passive after a modification that allows, in principle, a more practical implementation. II. PASSIVITY PROPERTIES OF THE NEGATIVE FEEDBACK HYPERBOLIC TANGENT COMPENSATOR Before proceeding with the study of the passivity properties of this scheme, it is important to remark that we are faced with infinite-dimensional delay-differential equations to which the standard tools cannot be applied directly. In [9] the authors show that, several reported statements of positive- PR discrete-time transfer functions are not completely correct, and then presented a lemma that gave the correct conditions for a system to be discrete-time passive, or equivalently discrete-time PR. This lemma is recalled here below for completeness, as well as a lemma taken from [1] which are the basis for our stability study. Lemma II.1 Discrete-time PR Consider an LTI discretetime system ykτ d + n D i1 D i ykτ d iτ d n N l N l ukτ d lτ d /8/$5. 8 IEEE 374

2 with τ d R +, k Z +, D l,n l R, n N n D. Assume the associated discrete-time transfer function nn He τ l N le lτ 1+ n D i1 D 1 ie iτ is discrete-time PR, that is, it satisfies i He τ is analytic in e τ >1. ii All poles of He τ on e τ 1are simple. iii Re{He jθ } for all θ R at which He jθ exists. iv If e jθ, θ R is a pole of He τ,andifr is the residue of He τ at e τ e jθ, then e jθ r. Then the system is discrete-time passive, that is, there exists β R such that N ykτ d ukτ d β k for all input sequences ukτ d L and all N Z + Lemma II. Passivity of continuous-time delayed syst. Consider an LTI continuous-time system described by the delay equation yt+ n D i1 D i yt iτ d n N l N l ut lτ d with τ d,t R +, D l,n l R, n N n D. Assume the discrete-time transfer function 1 is discrete-time PR. Then, the system is passive, that is, there exists β 1 IR such that t yτuτdτ β 1 for all input functions ut L and all t R +. The following expressions for the negative hyperbolic tangent compensator have been taken from [1] cosh ω tanh ω sinh ω e ω + e ω e ω e 1 e ω ω 1+e ω The block diagram of the previous expression involving a single delay line is presented in Fig. 1. This scheme is referred as the negative plus feedforward repetitive compensator. This repetitive scheme has the following equivalent expression that exhibits its infinite dimensionality. tanh ω ω k1 k1 s k ω s k 1 ω 3 Notice that, the poles are located at odd multiples of ω, while the zeros are located at the even multiples. Therefore, this scheme is also referred as the odd harmonics compensator. The frequency response is composed of resonant peaks of infinite magnitude located at the odd multiples of ω due to the poles, and notches at even multiples due to the zeros. Moreover, this scheme has also the following expression in the form of an infinite bank of harmonic oscillators tuned at odd harmonics of ω tanh ω ω π l1 4s s +l 1 ω The delay time required for the implementation of this scheme is given by τ d π ω, which will be used on the rest of this section to reduce the notation. Proposition II.3 The hyperbolic tangent scheme given by is discrete-time PR and thus passive. Proof: Rewriting in terms of the delay time τ d yiel He τ τd s tanh 1 e τ 1+e τ eτ 1 e τ The partial fraction expansion of this expression gives He τ 1 e τ hence the transfer function satisfies conditions i and ii of Lemma II.1. The residue associated with the fixed pole at e jθ 1 is r, and thus condition iv is satisfied. Finally, some simple computations prove that, Re{He jθ } Re{ j sinθ 1+cosθ }, thus fulfilling condition iii. This proves that the hyperbolic tangent scheme is discrete-time PR and, according to Lemma II., it is passive. In [1] a gain K is included as follows. 1 Ke τ 1+Ke τ 5 The aim of this practical modification is to prevent high gains in the resonance peaks and to enhance the robustness with respect to frequency variations. In fact, the peaks, originally of infinite magnitude, reach a maximum magnitude of 1+K while the notches reach a minimum magnitude of 1+K. This modification can also be seen as a frequency shifting process Hs Hs + a. Direct application of this shifting process to the exponential term results in e τ +a e τda e τ. In other wor, by proposing a gain factor K e τda we obtain 1 Ke τ 1+Ke τ 1 τ d s+a e τ+a 1+e e e τ d s+a τ +a e τ d s+a + e τ d s+a τd s+a sinh cosh τd s+a τd s + a tanh Therefore, if a gain K>1 is proposed, the poles and zeros move to the right, while if <K<1 is proposed then they move to the left. The following definition has been extracted from [11] and is used here to prove that the modified proposed scheme is strictly positive SPR

3 Definition II.4 A transfer function Gs is SPR if and only if there exists some ε> such that Gs ε is PR. Im Proposition II.5 The modified scheme 5 with <K<1 is SPR and thus strictly passive. 1+K 1+K 1+K Re Proof: According to Definition II.4, it should be proved that, there exists an ε> such that 1 Ke τ ε 1+Ke τ ε is positive {PR}. First, let us select ε a where a τ d lnk. Notice that, a> as far as <K<1. Second, consider K e τda, as defined above, which, after direct substitution, reduces expression 6 to tanh τ d s. The proof is completed by recalling that, tanh τ d s is PR according to Proposition II.3. Figure shows that, the Nyquist plot of scheme 5 K is a circle with radius and center 1+K, which is strictly contained in the right hand side of the complex plane Re{s}>. The last can be easily seen from the rectangular representation of tanh jωτ d given by jωτd tanh 1 Ke jωτ d 1+Ke jωτ d 1 K 1+K +K cosωτ d + j K sinωτ d 1+K +K cosωτ d Us - e ω negative feedforward Ys 3jω jω jω -jω -jω -3jω Fig. 1. Continuous-time model and poles/zeros location of the negative odd harmonics repetitive compensator with feedforward. Notice that, the maximum 1+K and minimum 1+K magnitudes occur whenever the part is zero, and subsequently, with zero phase shift, that is, at ωτ d l 1π and ωτ d lπ l 1,, 3,...,, respectively, which correspond to ω l 1ω odd multiples and ω lω even multiples. Figure 3 shows the frequency response of this modified scheme for different values of K. Notice that, the introduction of gain K does not cause a shifting of the resonance peaks nor of the notches. 6 Fig.. Nyquist plot of the negative odd harmonics compensator including a modification with a gain K, i.e., `1 Ke τ d s / `1+Ke τ. Mag [db] Phase [deg] K.95 K.75 K.5 K.95 K.75 K Frec [Hz] Fig. 3. Bode plot of the negative odd harmonics modified compensator, i.e., `1 Ke τ / `1+Ke τ for different values of gain K:.5,.75 and.95. III. PASSIVITY PROPERTIES OF THE POSITIVE FEEDBACK HYPERBOLIC COTANGENT REPETITIVE COMPENSATOR In what follows we consider the positive hyperbolic cotangent repetitive scheme, which is described by the following expressions taken from [1] cotanh ω cosh ω sinh ω e ω + e ω e ω e ω 1+e ω 1 e ω 7 Its block diagram is presented in Fig. 4. This scheme is referred as the positive plus feedforward repetitive compensator. This scheme has the following equivalent expression involving an infinite number of poles and zeros. s l1 l 1 cotanh ω ω l1 ω s l ω 8 Notice that, the poles located at every single multiple of ω, while the zeros are located exactly in the middle point between every two consecutive poles as shown in Fig. 4. Therefore, this scheme is also referred as all harmonics repetitive compensator. Similar to the odd harmonics compensator, damping can be added in the form of gain <K<1 to limit the resonance peak gains. The Bode plot for different values of K is shown 376

4 in Fig. 5. The Nyquist plot for the positive compensator is the same as for the negative compensator. See figure 6. feedforward jω 1.5j ω Im 1+K 1+K 1+K Re - e ω jω.5j ω -.5j ω Us positive Ys -jω -1.5j ω -jω Fig. 6. Nyquist plot of the positive all harmonics compensator including a modification with a gain K, i.e., `1+Ke τ d s / `1 Ke τ. Fig. 4. Continuos-time model and poles/zeros location of the positive all harmonics compensator with feedforward. feedforward Magnitude[dB] Phase[deg] K.95 K.75 K.5 K.95 K.75 K.5 Us - e 3ω - e 3ω Ys 7jω 6jω 5jω 3jω jω -jω -3j ω -5j ω -6j ω -7j ω Fig. 7. Continuous-time model and poles/zeros location of the 6l ± 1l, 1,, 3,... repetitive compensator with feedforward Fig. 5. Bode plot of the positive all harmonics modified compensator, i.e., `1+Ke τ / `1 Ke τ for different values of gain K:.5,.75 and.95.. In [1] it is shown that, Zs is PR if and only if 1/Zs is PR. And also that, Zs is SPR iff 1/Zs is SPR. According to this statements and based on the fact that cotanh 1/ tanh, it is straightforward to establish the validity of the following corollaries. The time delay required for the implementation of this scheme is given by τ d π ω, which will be used on the rest of this section to reduce the notation. Corollary III.1 The hyperbolic cotangent scheme given by 7 is PR and thus passive. Corollary III. The modified scheme 1+Ke τ 1 Ke τ 9 with <K<1 is SPR and thus strictly passive. In fact the modified scheme can also be written as cotanh τd s+a with a τ d lnk, and taking <K< 1 the poles and zeros are shifted to the left of the axis in the complex plane Re{s}<. IV. PASSIVITY PROPERTIES OF THE 6l ± 1 COMPENSATOR Finally, the 6l ± 1 l, 1,, 3,..., scheme is studied. This last compensator is described by the following expression Hs 1 e 3ω 1 1+e 3ω e 3ω Its block diagram is presented in Fig. 7. This scheme is referred as the 6l ± 1 repetitive compensator. The scheme has the following equivalent expression which involves an infinite number of poles and zeros. Hs e 3ω e 3ω e 3ω + e 3ω 1 3ω l1 s 3l ω sinh 3ω cosh 3ω 1 l s 6l ω where it is easy to see that the transfer function comprises an infinite number of poles located at ±j6lω and ±j6l 1ω l, 1,, 3,...,. Moreover, it also contains an infinite number of zeros located at ±j3lω. Proposition IV.1 The 6l±1 repetitive scheme given by 1 is discrete-time PR and thus passive. Proof: Rewriting 1 in terms of the time delay τ d used along this section to simplify the notation yiel π 3ω 377

5 He τ 1 e τ 1+e τ e τ e τ 1 e τ e τ 11 The partial fraction expansion of this expression gives He τ e τ 1+ e τ e τ i + 1 3i 1 e τ e iπ 3 e τ e iπ 3 Hence, the transfer function satisfies conditions i and ii of Lemma II.1. For iii it is found that Re{He jθ } Re{ jsinθ 1+ cosθ }. For the last condition, the two residues are given by r i and r 1 3 i, and the poles corresponding to each residue are θ 1 π 3 and θ π 3, respectively. Then e jθ1 r 1 1and e jθ r 1and the condition is fulfilled. This proves that, the 6l ± 1 scheme is discrete-time PR and, according to Lemma II., it is passive. Similar to previous cases, with the aim to prevent high gains in the resonance peaks and to enhance the robustness with respect to frequency variations, a gain K is included multiplying each delay line. This yiel the following modified expression He τ 1 K e τ 1+K e τ Ke τ 13 In fact, the peaks, originally of infinite magnitude, reach a maximum magnitude of 6K M 1 M K 4 + K K 14 and the notches reach either of the following two minimum magnitudes m 1 1 K 1+K + K 15 m 1 K 1 K + K 16 This modification can also be seen as a frequency shifting process of the form Hs Hs + a. Direct application of this shifting process to the exponential term results in τ e +a e τda e τ. In other wor, by proposing a gain factor K e τda we obtain 1 K e τ 1+K e τ Ke τ e τ+a e τ +a e τ +a + e τ +a 1 1 e τ +a 1+e τ +a e τ +a sinh τ d s + a cosh τ d s + a 1 Therefore, if a gain K>1 is proposed, the poles and zeros move to the right, while if <K<1 is proposed then they move to the left. Proposition IV. The modified scheme 13 with <K<1 is SPR and thus strictly passive. Proof: According to Definition II.4, it should be proved that, there exists an ε> such that 1 K e τ ε 1+K e τ ε Ke τ ε 17 is positive {PR}. First, let us select ε a where a τ d lnk. Notice that, ε a> as far as <K<1. Second, consider K e τda as defined above, which, after direct sinhτ substitution, reduces expression 17 to d s coshτ d s 1 sinhτ proof is completed by recalling that, d s is PR coshτ d s 1 according to Proposition IV K / m 1 K.5 m Fig K M 1 m m m M m M M 1 m 1 K.7 m M 1 M 1 3 Nyquist plot of the 6l ± 1l, 1,,.., harmonics compensator. Figure 8 shows that the Nyquist plot of scheme 13 goes from a flattened circle for <K< 1 to a cardioid for 1 <K< 3. Then, for 3 <K<1 the Nyquist plot becomes a limaon that approaches a circle of arbitrarily large radium as K gets closer to 1. It is clear that the range of interest lies in values of K slightly smaller than 1, i.e., when the Nyquist plot correspon to a limaon. Notice that, the Nyquist plot is strictly contained in the right hand side of the complex plane Re{s} >, as established by 15, that is, m 1 > and m > as far as <K<1. It is important to remark that, the maximum magnitudes in the resonance peaks does not occur exactly when the part equals zero, as a consequence, a small phase shift appears as it can be observed in Fig. 9. Another effect of the introduction of gain K is that ] the resonance peak occurs 3 K 1 at ω 6n±1ω ±[ π ω. which has been slightly shifted with respect to ω 6n±1ω. However, it should be noticed that this difference between the expected frequency, and the one obtained after introduction of gain K, ten to zero as K gets closer to one.

6 Magnitude[dB] Phase[deg] K.75 K.5 K.95 K.75 K.5 K Fig. 9. Bode plot of the modified 6l ± 1 compensator, i.e., `1 K e τ / `1+K e τ Ke τ for different values of gain K:.5,.75 and.95. [6] J. Ghosh and J. Paden. Nonlinear repetitive control, IEEE Trans. Automat. Contr., Vol. 45, No. 5,, pp [7] M. Tomizuka, T. Tsao and K. Chew. Discrete-time domain analysis and synthesis of repetitive controllers, In Proc. American Control Conf., pp , [8] R. Costa-Castelló and R. Griñó, A Repetitive Controller for Discrete-Time Passive Systems, Automatica, Vol. 4, 6, pp [9] C. Xiao, D. J. Hill, Generalization and new proof of the discretetime positive lemma and bounded lemma, IEEE Trans. on Circuits and Systems I, Vol 46, No. 6, June 1999, pp [1] A. de Rinaldis, R. Ortega et M. Spong, A compensator for attenuation of wave reflections in long cable actuator-plant interconnections with guaranteed transient performance improvement, Automatica, Vol. 4, No. 1, Oct 6, pp [11] J. H. Taylor, Strictly positive- functions and the Lefschetz- Kalman-Yakuvovich LKY lemma, IEEE Trans. on Circuits and Systems, March 1974, pp [1] P. Ioannou, G. Tao, Frequency domain conditions for strictly positive functions, Trans. on Automatic Control, Vol AC-3, No. 1, January 1987, pp [13] D. Rice, A Detailed Analysis of Six-Pulse Converter Harmonic Currents, IEEE Transactions on Industrial Electronics, Vol. 3, 1994, pp In the case of the zeros, all minimum magnitudes occur when the part is zero, and subsequently, with zero phase shift. Moreover, the minimums occur exactly at ωτ d lπ, which correspon to ω 3lω l, 1,,...,. V. CONCLUDING REMARKS This paper presented a study of the passivity properties of three repetitive schemes recently reported in literature. These schemes were referred as negative odd harmonics compensator, positive all harmonics compensator, and 6l ± 1 harmonics compensator. It was shown that all of them are discrete time positive, and thus passive. Moreover, it was shown that introducing a suitable damping, which turned out to be equivalent to a pole/zero shifting, all these schemes become strictly passive. Frequency responses are presented that showed the harmonic compensation capabilities of these schemes. VI. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of the Laboratory Franco-Mexican of Applied Control LAF- MAA through project MOPOFA-II. REFERENCES [1] G. Escobar, J. Leyva-Ramos and P. R. Martínez, Repetitive controllers with a feedforward path for harmonic compensation, IEEE Trans. on Ind. Electr., Vol. 54, Issue 1, Feb. 7, pp [] G. Escobar, P.G. Hernandez-Briones, R.E. Torres-Olguin, A.A. Valdez and M. Hernández-Gómez, A repetitive-based controller for the compensation of 6l ± 1 harmonic components, in Proc. IEEE International Symposium on Industrial Electronics ISIE, Vigo, Spain, June 4-7, 7. [3] S. Hara, T. Omata and M. Nakano. Synthesis of repetitive control of a proton synchrotron magnet power supply, in Proc. 8th World Congr. IFAC, Vol. XX, pp , [4] S. Hara, T. Omata and M. Nakano. Synthesis of repetitive control systems and its applications, in Proc. 4th Conf. Decision and Control, Vol. 3, pp , [5] S. Hara, Y. Yamamoto, T. Omata and M. Nakano. Repetitive control systems: A new type servo systems and its applications, IEEE Trans. Automat. Contr., Vol. 33, No. 7, 1988, pp